Decoding a Class of Affine Variety Codes with Fast DFT

TL;DR

This paper presents a fast decoding method for affine variety codes using multidimensional DFT and Grobner basis, improving error-value calculation efficiency in erasure and error correction tasks.

Contribution

It introduces a novel decoding procedure combining multidimensional DFT with Grobner basis-based recurrence relations for affine variety codes.

Findings

Reduces computational complexity of error-value calculations.

Effective for Reed-Solomon and Hermitian codes.

Enhances decoding efficiency in affine variety codes.

Abstract

An efficient procedure for error-value calculations based on fast discrete Fourier transforms (DFT) in conjunction with Berlekamp-Massey-Sakata algorithm for a class of affine variety codes is proposed. Our procedure is achieved by multidimensional DFT and linear recurrence relations from Grobner basis and is applied to erasure-and-error decoding and systematic encoding. The computational complexity of error-value calculations in our algorithm improves that in solving systems of linear equations from error correcting pairs in many cases. A motivating example of our algorithm in case of a Reed-Solomon code and a numerical example of our algorithm in case of a Hermitian code are also described.